if the coordinates of the mid point of the sides BC,CA and AB of a triangle abc are (3,4)(4,6) and (5,7) respectively.find the vertics of triangle abc
Answers
Answer:
Let A(x
1
,y
1
), B(x
2
,y
2
) and C(x
3
,y
3
) be the vertices of △ABC. Let D(1, 2), E(0, -1), and F(2, -1) be the mid-points of sides BC, CA and AB respectively
Since D is the mid-point of BC
∴
2
x
2
+x
3
=1and
2
y
2
+y
3
=2
⇒x
2
+x
3
=2andy
2
+y
3
=4
Similarly, E and F are the mid-points of CA and AB respectively.
∴
2
x
1
+x
3
=0and
2
y
1
+y
3
=−1
⇒x
1
+x
3
=0andy
1
+y
3
=−2
and,
2
x
1
+x
2
=2and
2
y
1
+y
2
=−1
⇒x
1
+x
2
=4andy
1
+y
2
=−2
From (i), (ii) and (iii), we get
(x
2
+x
3
)+(+x
1
x
3
)+(x
1
+x
2
)=2+0+4and,(y
2
+y
3
)+(y
1
+y
3
)+(y
1
+y
2
)=4−2−2
⇒2(x
1
+x
2
+x
3
)=6and2(y
1
+y
2
+y
3
)=0
⇒x
1
+x
2
+x
3
=3andy
1
+y
2
+y
3
=0
From (i) and (Iv), we get
x
1
+2=3andy
1
+4=0
⇒x
1
=1andy
1
=−4
So, the coordinates of A are (1,-4)
From (ii) and (iv), we get
x
2
+0=3andy
2
−2=0
⇒x
2
=3andy
2
=2
So, coordinates of B are (3, 2)From (iii) and (iv), we get
x
3
+4=3andy
3
−2=0
⇒x
3
=−1andy
3
=2
So, coordinates of C are (-1, 2)
Hence, the vertices of the triangle ABC are A(1, -4), B(3, 2) and (-1, 2).